Subagent report: stage-1 falsifiers (2026-08-11)

Provenance: Claude Opus subagent, dispatched for stage 1 only (census, exact-gap ladder, invariant suite) per SPEC-addendum. Report reproduced verbatim below. Fable review: construction validated (100/100 tests, crystallographic anchor, two-to-three-method exact count agreement, the quotient-graph trap caught and closed); the census finding — frozen states in W=0 at every cell, persisting to arbitrary size by tiling — is REAL and changes the project’s framing: all gap/τ quantities are giant-component quantities from here on. Also corrects the SPEC-addendum’s ladder estimates (bulk-entropy Pauling estimates undercount thin-cell sectors by 3.5–49×).

Verified to 12 significant figures. Stage 1 is complete.

Report — Stage 1 falsifiers, ice-mixing

Bottom line: no falsifier fired against the construction, but the census found a real and permanent obstruction — the chain is reducible on the full W=0 sector at every system size. All artefacts are at ~/repo/project/ice-mixing/. Git untouched; no venv needed (numpy 1.26.4 / scipy 1.17.1 / pytest 9.0.2 already present).

File inventory

Actual vs estimated sector sizes (measured first, as required)

The addendum’s ladder estimates are substantially too low — thin cells carry far more entropy per spin than the bulk 0.2027, and the excess grows as the cell thins. Independent methods: DFS enumeration, frontier/transfer-matrix DP (counts and the joint winding histogram, never materialising a state), and brute-force 2^16 at the smallest cell. All agree exactly, including full joint W-histograms. External anchor: (1,1,1) reduces to K(4,4), whose ice states are the 4×4 zero-one matrices with row/column sums 2 — the classical 90, reproduced. Axis permutations (1,1,3)↔︎(3,1,1) and (1,2,2)↔︎(2,2,1) agree. (2,2,2) skipped with reason (DP blows past 10.5M frontier keys; ~2e11 states).

Census — flag this prominently

1. Frozen states exist in W=0 at every cell examined, and they persist to arbitrarily large cells. Periodically tiling a small-cell frozen state gives, verified from scratch in the target cell’s own hexagon set, a frozen W=0 ice state of (2,2,2), (2,2,4), (1,2,4) and the 432-spin (3,3,3) — 10/10 cases. So the full W=0 sector is never connected, Δ=0 on it at every L, and every gap/τ in this project is a giant-component quantity. Density falls with size (0.092% at (1,2,2)), so it is measure-zero asymptotically but real at finite L.
2. Disconnection is an aspect-ratio effect, not a chain defect. Quasi-1D (1,1,c) cells shatter; the one cell with a 2-D cross-section, (1,2,2), has a 99.91% giant component with only isolated frozen states outside it.
3. Singleton components are exactly the frozen states in every case; no component ever straddles two winding sectors (checked over all 25/69/137 sectors at the three smallest cells). Global arrow reversal acts as a permutation of components and explains the equal-size component pairs.
4. Caveat: (1,2,2) is the only non-quasi-1D cell reachable by enumeration. Stage 1 says nothing about connectivity in the cubic geometry stage 3 will use.

Exact gaps (all restricted to the giant component — stated loudly)

Sanity Δ>0 iff connected: holds at all five cells (full-sector Δ = 0 to machine precision everywhere). Lanczos vs dense agree to 0 in double precision where both ran; the (1,2,2) headline was re-verified by 60,000 power iterations on the deflated operator — agrees to 12 significant figures. (1,2,3) and (1,1,5): counted but not enumerated (≈4 h / 40 min DFS plus tens of GB), no gap — skipped with reason. These anisotropic anchors are not a z measurement.

Invariants — all clean

• Ice rule + winding after 4,369,925 applied moves across five cells: 0 failures (64-spin cells capped at the first 200k states for budget).
• Flip is an involution preserving its hexagon’s flippability: 0 failures. Fast 6-bit predicate vs a definition-level reference walk: 0 mismatches.
• P exactly symmetric and doubly stochastic at all five cells ⇒ exact detailed balance w.r.t. uniform. Empirical χ² on samples thinned by 20 exact relaxation times: p = 0.250 / 0.632 / 0.518; detailed-balance max |z| = 1.75 / 2.59 / 2.89.
• Trajectory at (1,2,2): 100k attempts, 8,678 moves, 0 ice/winding failures.

Anomalies worth the supervisor’s attention

1. The quotient-graph trap. Enumerating “hexagons” as 6-cycles of the periodic quotient graph is wrong on small cells: at (1,1,1) that graph is K(4,4) with 96 6-cycles versus 16 real hexagons. All 80 extras are non-contractible, stay within the ice rule, and change W in 80/80 tested cases — exactly the silent flux-sector corruption the brief warned about. The implementation derives hexagons from adjacency plus displacement, and the derived counts (6 per bond, 12 per site, n_hex = n_bonds = 16abc) are asserted in tests alongside the hand derivation.
2. (1,1,1)’s W=0 sector is entirely frozen (12 states, 12 components). Not a bug; the connected components at that cell all live in W=±2 sectors.
3. Crystallographic anchor passed on four shapes ((1,1,1), (1,1,2), (1,2,2), (2,2,2)): constructive bijection, identical edge sets, degree 6, tetrahedra = diamond sites, matching adjacency spectra. A deliberate single-bond corruption is confirmed to break it.

Timings

Full run 283 s wall (CPU only). Slowest items: DFS at (1,2,2) 32 s and (1,1,4) 45 s; axis-permutation DP checks 145 s; Lanczos ≤ 3 s. Test suite 184 s. Nothing exceeded the 10-minute per-computation budget.

Recommendations for stage 2 (not started, per the gate)

1. Start every chain inside the giant component and record the convention — a uniformly drawn W=0 state is unreachable 93% of the time at (1,1,4).
2. Measure the giant-component fraction at cubic L=2 empirically before the campaign; enumeration cannot reach it.
3. Do not use the (1,1,c) gaps as z anchors.
4. Carry Δ in explicit units — attempt vs sweep differs by n_hex = 16abc.